Math 223, Section 101—Homework #2
due Wednesday, September 21, 2011 at the beginning of class
Practice problems. Don’t turn in these problems. Most of these problems are easy, and most of
them have answers in the back of the book, so they will be useful to you to check your understanding of the material.
1. Let x and y be nonzero vectors in a vector space V , and let a and b be nonzero real numbers.
Show that:
(a) if ax = 0, then a = 0 or x = 0.
(b) if ax = ay, then x = y.
(c) if ax = bx, then a = b. In other words, all scalar multiples of a nonzero vector x are
different from one another.
2. Friedberg, Insel, and Spence, Section 1.4, p. 33, #2(a),(c),(e), #3(a),(c),(e), and #4(a),(c),(e)
3. Friedberg, Insel, and Spence, Section 1.4, p. 34, #5(a),(c),(e),(g)
4. Friedberg, Insel, and Spence, Section 3.4, pp. 194–195, #1(d),(e)
5. Friedberg, Insel, and Spence, Section 3.4, pp. 195–196, #2(a),(c),(e),(g),(i)
6. Friedberg, Insel, and Spence, Section 1.5, p. 40, #1(a),(b),(d)–(f)
7. (a) Suppose the coefficient matrix of a system of three linear equations in three variables
has a pivot in each column. Is the system consistent or inconsistent? If consistent, is
the solution unique?
(b) Suppose the augmented matrix of a system of four linear equations in three variables
has a pivot in each column. Is the system consistent or inconsistent? If consistent, is
the solution unique?
(c) Suppose the augmented matrix of a system of three linear equations in three variables
has a pivot in each column except the last column. Is the system consistent or inconsistent? If consistent, is the solution unique?
8. Friedberg, Insel, and Spence, Section 1.5, pp. 40–41, #2(a),(c),(e),(g),(i)
9. Friedberg, Insel, and Spence, Section 1.5, p. 41, #3–5. In #4, replace F n by Rn ; in #5,
replace Pn (F ) by Pn (R).
10. Friedberg, Insel, and Spence, Section 1.5, p. 41, #7. There are many possible answers;
come up with at least one answer different from the one in the back of the book.
11. Friedberg, Insel, and Spence, Section 1.5, p. 42, #12 and #18
Homework problems. Write up and hand in solutions to these problems. Please have your solutions to these problems in the correct order on your pages. Write clearly and legibly, in complete
sentences; the solutions you hand in will be better if you think about how to best phrase your answer before you begin to write it. Please staple the pages together (unstapled solutions will not be
accepted).
1. Friedberg, Insel, and Spence, Section 3.4, p. 195, #2(b),(d),(f). Don’t be afraid if you get
double-digit denominators in (d).
(continued on next page)
2. Friedberg, Insel, and Spence, Section 1.4, p. 34, #5(d),(f),(h)
3. You are given a system of three linear equations in three variables. Let A be the coefficient
matrix of the system, so that A is a 3 × 3 matrix.
(a) Suppose that the entries in each row of A add to zero. Prove that your system either is
inconsistent or else has infinitely many solutions.
(b) Suppose that the entries in each column of A add to zero. Prove that your system
either is inconsistent or else has infinitely many solutions.
4. In the vector space P (R), consider the subset S = {1 − x, x − x2 , x2 − x3 , . . . }. Notice
the telescoping infinite series
1 = (1 − x) + (x − x2 ) + (x2 − x3 ) + · · · .
Despite this, show that 1 is not in span(S).
5. Friedberg, Insel, and Spence, Section 1.5, pp. 40–41, #2(d),(f),(h)
6. Friedberg, Insel, and Spence, Section 1.5, p. 41, #9 and #10
7. Let V be a vector space, and let u, v, w be distinct vectors in V . Show that the set {u, v, w}
is linearly independent if and only if the set {u + v, v + w, w + u} is linearly independent.
8. Let r1 , . . . , rn be distinct real numbers, and define the following functions in F(R, R):
f1 (t) = er1 t ,
f2 (t) = er2 t ,
...,
fn (t) = ern t .
Show that {f1 (t), . . . , fn (t)} is a linearly independent set. (Suggestion: use your knowledge of calculus, and consider what happens as t → ∞.)
Bonus question. Given a polynomial p(x) that is not the zero polynomial and a real number a,
define the order of vanishing of p(x) at x = a to be the nonnegative integer k such that p(a) = 0,
p0 (a) = 0, . . . , and p(k−1) (a) = 0, but p(k) (a) 6= 0. For example:
• The order of vanishing of p(x) = mx + b is 1 at x = −b/m and 0 everywhere else.
• The order of vanishing of p(x) = x7 (x − 2)9 is 7 at x = 0 and 9 at x = 2 (and 0 elsewhere).
Let the polynomials p1 (x), . . . , pn (x) ∈ P (R) \ {0} have orders of vanishing k1 , . . . , kn , respectively, at x = a. Prove that if k1 , . . . , kn are all distinct, then {p1 (x), . . . , pn (x)} is linearly independent.